= Clarkson's inequalities
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= Clarkson inequality
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For $2\leq p<\infty$, the Clarkson inequality is
$$
\left\lVert\frac{f+g}{2}\right\rVert_p^p
+\left\lVert\frac{f-g}{2}\right\rVert_p^p
\leq\frac{\lVert f\rVert_p^p+\lVert g\rVert_p^p}{2}.
$$
For $1<p\leq2$ and $q=p/(p-1)$, the other <Clarkson inequality> is
$$
\left\lVert\frac{f+g}{2}\right\rVert_p^q+\left\lVert\frac{f-g}{2}\right\rVert_p^q\leq\left(\frac{\lVert f\rVert_p^p+\lVert g\rVert_p^p}{2}\right)^{q/p}.
$$
Together these inequalities prove uniform convexity of $L^p$ for $1<p<\infty$.
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